Division algebras with infinite genus

Abstract

We give an explicit construction of division algebras with infinite genus. We go on to show that there exists a field for which (1) there are infinitely many nonisomorphic quaternion division algebras with center, and (2) any two quaternion division algebras with center are pairwise weakly isomorphic. In fact, we show that there are infinitely many nonisomorphic fields satisfying these two conditions. © 2014 London Mathematical Society.